Nina Zorboska

Toeplitz operators with BMO symbols and the Berezin transform

We prove that the boundedness and compactness of the Toeplitz operator on the Bergman space with a BMO1 symbol is completely determined by the boundary behaviour of its Berezin transform. This result extends the known results in the cases when the symbol is either a positive L1-function or an L∞ function.

Composition operators with closed range

We characterize the closed-range composition operators on weighted Bergman spaces in terms of the ranges of the inducing maps on the unit disc. The method uses Nevanlinnas counting function and Lueckings results on inequalities on Bergman spaces.

Composition operators on weighted Dirichlet spaces

We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

Sampling sets and closed range composition operators on the Bloch space

We give a necessary and sufficient condition for a composition operator C Φ on the Bloch space to have closed range. We show that when Φ is univalent, it is sufficient to consider the action of C Φ on the set of Mobius transforms. In this case the closed range property is equivalent to a specific sampling set satisfying the reverse Carleson condition.

The Berezin transform and radial operators

We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L 1 (D) symbol.

Toeplitz operators on Bloch-type spaces

We characterize complex measures μ on the unit disk for which the Toeplitz operator T α μ , a > 0, is bounded or compact on the Bloch type spaces B α .

Univalently Induced, Closed Range, Composition Operators on the Bloch-type Spaces

While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces Bα with α 6= 1 are the ones induced by a disc automorphism. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2 e-mail: zorbosk@cc.umanitoba.ca Received by the editors May 22, 2009; revised August 17, 2009. Published electronically April 6, 2011. Research supported in part by NSF grant, DMS 0200587. AMS subject classification: 47B35, 32A18.

Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

We determine the boundedness and compactness of a large class of operators, mapping from general Banach spaces of holomorphic functions into a particular type of spaces of functions determined by the growth of the functions, or the growth of the functions derivatives. The results show that the boundedness and compactness of such intrinsic operators depends only on the behaviour on the kernel functions. They also generalize previous similar results about several specific classes of operators, such as the multiplication, composition and integral operators.

Isometric and Closed-Range Composition Operators between Bloch-Type Spaces

We present an overview of the known results describing the isometric and closed-range composition operators on different types of holomorphic function spaces. We add new results and give a complete characterization of the isometric univalently induced composition operators acting between Bloch-type spaces. We also add few results on the closed-range determination of composition operators on Bloch-type spaces and present the problems that are still open.

Uniform local univalence and hyperbolic distortion

We show that for , an analytic self-map of the unit disk with bounded weighted hyperbolic local distortion, the distortions boundedness from below on a subdomain implies that is uniformly locally univalent on .